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Distribution for nonsymmetric V-monotone position operators

Adrian Dacko, Lahcen Oussi

Abstract

We investigate the vacuum distribution of a family of partial sums of nonsymmetric position operators, depending on a real parameter $λ$, and acting on the discrete Fock space in the framework of V-monotone independence. We analyze the combinatorics of the moments of this distribution, and using its Cauchy--Stieltjes transform, we determine its exact form, consisting of a unique atom and an absolutely continuous part. Finally, we present computer-generated graphs that illustrate the distribution for several values of the intensity parameter $λ$.

Distribution for nonsymmetric V-monotone position operators

Abstract

We investigate the vacuum distribution of a family of partial sums of nonsymmetric position operators, depending on a real parameter , and acting on the discrete Fock space in the framework of V-monotone independence. We analyze the combinatorics of the moments of this distribution, and using its Cauchy--Stieltjes transform, we determine its exact form, consisting of a unique atom and an absolutely continuous part. Finally, we present computer-generated graphs that illustrate the distribution for several values of the intensity parameter .
Paper Structure (3 sections, 5 theorems, 21 equations)

This paper contains 3 sections, 5 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Combinatorics of noncrossing partitions and V-monotone labelings
  3. Discrete V-monotone Fock space and related operators

Key Result

Theorem 1.2

Let $\{ a_{N,i} : N \in \mathbb{N}_{+}, i \in [N] \}$ be a family of operators with the following properties: Then where $\gamma(\lambda)$ is a random variable with the Poisson distribution with intensity $\lambda > 0$.

Theorems & Definitions (11)

  • Definition 1.1
  • Theorem 1.2
  • Definition 3.1: Discrete V-monotone Fock space
  • Lemma 3.2
  • Proposition 3.3
  • Definition 3.4
  • Proposition 3.5
  • proof
  • Remark 3.6
  • Proposition 3.7
  • ...and 1 more